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Metamath Proof Explorer

Theorem cdleme9tN

Description: Part of proof of Lemma E in Crawley p. 113, 2nd paragraph on p. 114. X and F represent t_1 and f(t) respectively. In their notation, we prove f(t) \/ t_1 = q \/ t_1. (Contributed by NM, 8-Oct-2012) (New usage is discouraged.)

Ref Expression
Hypotheses cdleme9t.l ⊢ ≤ = ( le ‘ 𝐾 )
cdleme9t.j ⊢ ∨ = ( join ‘ 𝐾 )
cdleme9t.m ⊢ ∧ = ( meet ‘ 𝐾 )
cdleme9t.a ⊢ 𝐴 = ( Atoms ‘ 𝐾 )
cdleme9t.h ⊢ 𝐻 = ( LHyp ‘ 𝐾 )
cdleme9t.u ⊢ 𝑈 = ( ( 𝑃 ∨ 𝑄 ) ∧ 𝑊 )
cdleme9t.g ⊢ 𝐹 = ( ( 𝑇 ∨ 𝑈 ) ∧ ( 𝑄 ∨ ( ( 𝑃 ∨ 𝑇 ) ∧ 𝑊 ) ) )
cdleme9t.x ⊢ 𝑋 = ( ( 𝑃 ∨ 𝑇 ) ∧ 𝑊 )
Assertion cdleme9tN ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ 𝑄 ∈ 𝐴 ∧ ( 𝑇 ∈ 𝐴 ∧ ¬ 𝑇 ≤ 𝑊 ) ) ∧ ¬ 𝑇 ≤ ( 𝑃 ∨ 𝑄 ) ) → ( 𝐹 ∨ 𝑋 ) = ( 𝑄 ∨ 𝑋 ) )

Proof

Step Hyp Ref Expression
1 cdleme9t.l ⊢ ≤ = ( le ‘ 𝐾 )
2 cdleme9t.j ⊢ ∨ = ( join ‘ 𝐾 )
3 cdleme9t.m ⊢ ∧ = ( meet ‘ 𝐾 )
4 cdleme9t.a ⊢ 𝐴 = ( Atoms ‘ 𝐾 )
5 cdleme9t.h ⊢ 𝐻 = ( LHyp ‘ 𝐾 )
6 cdleme9t.u ⊢ 𝑈 = ( ( 𝑃 ∨ 𝑄 ) ∧ 𝑊 )
7 cdleme9t.g ⊢ 𝐹 = ( ( 𝑇 ∨ 𝑈 ) ∧ ( 𝑄 ∨ ( ( 𝑃 ∨ 𝑇 ) ∧ 𝑊 ) ) )
8 cdleme9t.x ⊢ 𝑋 = ( ( 𝑃 ∨ 𝑇 ) ∧ 𝑊 )
9 1 2 3 4 5 6 7 8 cdleme9 ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ 𝑄 ∈ 𝐴 ∧ ( 𝑇 ∈ 𝐴 ∧ ¬ 𝑇 ≤ 𝑊 ) ) ∧ ¬ 𝑇 ≤ ( 𝑃 ∨ 𝑄 ) ) → ( 𝐹 ∨ 𝑋 ) = ( 𝑄 ∨ 𝑋 ) )